In this work we analyze error estimates for rational approximation methods, and their stabilizations, for strongly continuous semigroups. Chapter 1 consists of a brief survey of time discretization methods for semigroups. In Chapter 2, we demonstrate a new method for obtaining convergent approximations in the absence of stability for strongly continuous semigroups with arbitrary initial data. In Section 2.2, we state the stabilization result in more general form and show that this method can be used to improve known error estimates by a magnitude of up to one half for smooth initial data. In Section 2.3, we give concrete examples of some of these stabilizers. Section 2.4 concerns abstract stabilization results, including stabilized Trotter-Kato and Lax-Chernoff theorems. In Chapter 3, we use numerical quadrature formulas for Banach space valued functions in order to approximate semigroups that can be represented via

the Hille-Phillips functional calculus. In particular, we find error estimates for our

approximation method for the semigroup generated by the square root of a semigroup